**14**

votes

**0**answers

447 views

### Maximal Tori and group structures on spheres

It is known that for any compact Lie group $G$ with maximal torus $T$, that any other maximal torus $T'$ is conjugate to $T$. This might be a bit of a stretch, but I was wondering if it is possible to ...

**14**

votes

**0**answers

930 views

### What groups are Lie groups?

We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...

**13**

votes

**0**answers

416 views

### To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...

**12**

votes

**0**answers

495 views

### How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...

**12**

votes

**0**answers

306 views

### Nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...

**11**

votes

**0**answers

470 views

### Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary,
the result of historical accidents. In order to avoid repetitions, the four infinite
families $A_\ell, B_\ell, ...

**11**

votes

**0**answers

450 views

### Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?

To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$
(vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching
an extra vertex to every old vertex in $Q_0$. Then ...

**10**

votes

**0**answers

221 views

### differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...

**10**

votes

**0**answers

529 views

### Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...

**10**

votes

**0**answers

425 views

### How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...

**9**

votes

**0**answers

292 views

### Geometrizing the Third Cohomology of a Complex Lie Group

If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural ...

**9**

votes

**0**answers

346 views

### Reflection groups in O(n+1,n) arising `in nature'?

For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...

**9**

votes

**0**answers

243 views

### Does a symplectic group act on tensor power of spin representation?

More specifically, let $S_k$ be the spin representation of $\mathrm{Spin}(2k+1)$.
Then is there are action of $\mathrm{Sp}(2r-2)$ on $\otimes^{2r}S_k$ which commutes
with the action of ...

**8**

votes

**0**answers

384 views

### Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...

**8**

votes

**0**answers

271 views

### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

**8**

votes

**0**answers

359 views

### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

**7**

votes

**0**answers

134 views

### $v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...

**7**

votes

**0**answers

85 views

### Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...

**7**

votes

**0**answers

280 views

### subgroups of higher rank lattices

This is related to the question $G=\langle a\rangle H$ for subgroup $H$ raised a few days ago. Suppose $\Gamma $ is a higher rank lattice (for example, $SL_3({\mathbb Z})$). As Misha says in his ...

**7**

votes

**0**answers

151 views

### Reference Request - Spaces of Smooth Vectors

I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...

**7**

votes

**0**answers

404 views

### Is endoscopy interesting in simply-laced cases?

Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...

**7**

votes

**0**answers

490 views

### Small sum of group elements acting as rank 1 matrix.

I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some ...

**7**

votes

**0**answers

353 views

### For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?

The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...

**6**

votes

**0**answers

241 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**6**

votes

**0**answers

130 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

**6**

votes

**0**answers

287 views

### Injectivity of Lie group exponential function

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...

**6**

votes

**0**answers

144 views

### Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...

**6**

votes

**0**answers

351 views

### Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...

**6**

votes

**0**answers

309 views

### Must the adjoint representation of a semisimple Lie group be algebraic?

Suppose G is a real semisimple Lie group without compact factor, then Aut(g) is obviously an algebraic subgroup of SL(n,R) where g is the Lie algebra and n is the dimension. Regard the adjoint ...

**6**

votes

**0**answers

215 views

### Cohomology of T^{n}/W for compact Lie groups

Let $G$ be a compact, connected and simply connected.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ for $n\ge ...

**6**

votes

**0**answers

566 views

### Baker-Campbell-Hausdorff formula: prime divisors of denominators

Consider the Baker-Campbell-Hausdorff formula (Wikipedia page):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dots$$
Many sources, including the Wikipedia ...

**5**

votes

**0**answers

112 views

### Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...

**5**

votes

**0**answers

109 views

### Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...

**5**

votes

**0**answers

78 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**5**

votes

**0**answers

177 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**5**

votes

**0**answers

109 views

### Explicit generators for homotopy groups of Lie groups

I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$.
It is ...

**5**

votes

**0**answers

378 views

### A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?

The following is my first question here on mathoverflow.
Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...

**5**

votes

**0**answers

212 views

### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...

**5**

votes

**0**answers

412 views

### Connections on a Lie Group

A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...

**5**

votes

**0**answers

133 views

### Which cocompact subgroups of $G$ do contain a cocompact normal subgroup of $G$?

Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$.
Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of ...

**5**

votes

**0**answers

243 views

### Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...

**5**

votes

**0**answers

134 views

### Is there an analogue of spin/oscillator representation for the general linear Lie algebra?

(Work over complex numbers)
Let $V$ be an orthogonal space. Let $Pin(V)$ be the double cover of the orthogonal group $O(V)$. Then $Pin(V)$ has a basic spin representation which we can think of as the ...

**5**

votes

**0**answers

216 views

### Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...

**5**

votes

**0**answers

354 views

### Computing centralizers in Lie groups

Let $G$ be a real semisimple Lie group. Really, I only care about $\text{SL}(n,\mathbb{R})$ and $\text{Sp}(2n,\mathbb{R})$.
I'd like to perform a computer search for a finite group with a certain ...

**5**

votes

**0**answers

171 views

### average Riemannian distance between Identiity and a random point in SO(n) or SU(n)

I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...

**5**

votes

**0**answers

444 views

### Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...

**4**

votes

**0**answers

246 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**4**

votes

**0**answers

82 views

### Connected compact Lie groups with Lie algebra so(4n, R)

I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case.
Consider the Lie ...

**4**

votes

**0**answers

100 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

**4**

votes

**0**answers

186 views

### Decompositions of a compact Lie group into “fixed point set types”

Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...